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UC-NRLF 

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LIBRARY 

OF  THE 

University  of  California. 

GIFT  OF 


On  the  Representation  of  a  Function  by  a 
Trigonometric  Series. 


DISSERTATION 


PRESENTED   TO  THE  BOARD  OF  UNIVERSITY  STUDIES  OF  THE 

JOHNS  HOPKINS  UNIVERSITY  FOR  THE  DEGREE 

OF  DOCTOR  OF  PHILOSOPHY 


EDWARD  !P.  MANNING. 

ii 


BALTIMORE,  1894. 


nr 


PRESS   OF 

THE   FRIEDENWALD   COMPANY, 

BALTIMORE. 


-M3 


PREFACE. 

In  this  thesis  I  have  considered  the  representation,  in  trigonometric  series, 
of  a  function  of  a  real  variable  only. 

Since  this  subject  has  already  been  so  fully  treated  I  could  hardly  expect 
to  obtain  many  new  results.  Accordingly  it  has  been  my  special  aim,  after 
carefully  examining  the  work  of  others,  to  investigate  independently  certain 
phases  of  the  question,  and  thereby  to  obtain  if  possible  a  simpler  development 
of  the  subject.  To  some  extent  my  efforts  have  been  rewarded,  since,  in  several 
cases,  I  have  arrived  at  results  by  methods  considerably  shorter  than  those 
which  others  have  employed. 

The  first  two  sections  of  this  paper  are  introductory.  Section  1  is  purely 
historical,  giving  a  brief  account  of  the  origin  of  the  question  and  an  outline  of 
the  principal  work  done  upon  it  up  to  the  present  time.*  In  section  2  is  pre- 
sented the  theorem  of  du  Bois-Eeymond  which  gives  directly  the  form  of  the 
coefficients  in  the  trigonometric  series  and  proves  for  all  cases  the  uniqueness 
of  the  development. 

In  the  next  two  sections  I  have  shown  that  the  convergence  of  this  develop- 
ment of  an  integrable  function  /  (x)  to /(#„),  for  x  =  x0,  depends  only  upon  the 
behavior  of  the  function  in  the  vicinity  of  x0 .  This  is  followed  in  sections  5 
and  6  by  a  proof  that  the  series  thus  converges  tof(x0): 

1°.  When  f(x),  in  the  neighborhood  of  x0>  is  finite,  possesses  only  a  finite 
number  of  discontinuities  and  only  a  finite  number  of  maxima  and  minima. 

2°.  When  it  satisfies  the  very  general  condition 

limr-|/(*o±2r)-,/>t±o)| 

<=oJo  y 

This  condition  I  have  compared  with  other  conditions  and  have  shown  its 
application  to  an  important  class  of  functions  in  modern  analysis,  viz.,  to  func- 
tions having  an  infinite  number  of  maxima  and  minima,  or  an  infinite  number 
of  discontinuities  in  a  finite  region.  In  section  7  I  have  given  an  illustration 
of  a  function  of  this  kind  whose  trigonometric  development  diverges  for  one 

*For  a  more  complete  historical  treatment  see  the  papers  of  Riemann  (Math.  Werke, 
p.  213 ;  found  also  in  Bulletin  des  Sci.  Math.,  Vol.  V,  1873,  p.  20),  Sachse  (Bulletin  des  Sci. 
Math.,  1880,  p.  43),  and  Gibson  (Proceedings  of  the  Edinburgh  Math.  Society,  Vol.  XI,  p.  137.) 


111.2 


value  of  the  variable.  Finally,  in  the  last  section,  I  have  discussed  the  nature 
of  the  convergence  of  the  series,  treating  especially  the  question  of  uniform  con- 
vergence. 

I  desire  here  to  express  my  acknowledgment  to  Professor  Craig,  who  sug- 
gested to  me  the  subject  of  this  thesis  and  who  has  had  general  supervision 
over  its  preparation,  and  also  to  Professor  Franklin,  who,  by  his  advice  and 
suggestions,  has  afforded  me  valuable  assistance  on  several  points. 

Baltimore,  April  3,  1894. 


CONTENTS. 

Section  Page 

1.  Historical  account  of  the  origin  and  development  of  the  question,        .       1 

2.  Du  Bois-Reymond's  theorem  which  gives  the  form  of  the  coefficients 

in  the  trigonometric  series  and  shows  that  the  development  is 
unique,      ...........       6 

3.  The  question  of  the  convergence  of  the  series,  leading  to  Dirichlet's 

integral,      .         .         .         .         .         .         .         .         .         .         .12 

4.  Proof  that  the  convergence  of  the  series  to  the  value  of  the  function, 

for  any  particular  value  of  the  variable,  depends  only  upon  the 
nature  of  the  function  in  the  vicinity  of  that  value,      .         .         .     14 

5.  Proof  that  the  development  is  applicable  to  functions  satisfying  Dirich- 

let's  conditions, 16 

6.  Derivation  of  the  condition 

lim f'lpfr)  —  y(+Q)l  dy _ o, 


lo  r 

and  illustrations  of  its  applicability,    . 

7.  Example  of  a  function  not  developable  at  every  point, 

8 .  Discussion  of  the  nature  of  the  convergence  of  the  series, 
Biographical  Sketch, 


18 
21 
24 
27 


ON  THE  REPRESENTATION  OF  A  FUNCTION   BY  A 
TRIGONOMETRIC  SERIES. 


1.  The  question  as  to  the  possibility  of  representing  an  arbitrary  function 
of  a  real  variable  by  means  of  a  trigonometric  series,  first  suggested  nearly  a 
century  and  a  half  ago,  has  received  considerable  attention.  Much  has  been 
written  upon  it,  for  it  is  important,  not  only  in  pure  analysis,  but  also  in  the 
practical  applications  of  mathematics  to  problems  in  physics  and  astronomy. 

The  question  arose  from  a  comparison  of  two  different  forms  of  solution 
of  the  partial  differential  equation  for  a  vibrating  chord,  viz : 

a<2— c  dx2  [ } 

Integrals  of  this  equation  were  obtained  in  the  form  of  a  general  solution 
containing  an  arbitrary  function  and  in  the  form  of  a  trigonometric  series. 
The  general  solution,  subjected  to  the  condition  that  the  extremities  of  the 
chord  are  at  rest,  becomes 

?==/(« +  C*)—  f(ct  —  »), 

where  f  is  an  arbitrary  function  having  for  a  period  the  double  length  of  the 
chord.  This  general  solution  was  first  obtained  by  d'Alembert,*  who,  however, 
supposed  that,/ was  of  such  a  nature  that  it  could  always  be  represented  by  a 
continuous  curve,  this  being  the  idea  conveyed  to  mathematicians  of  that  time 
by  the  expression  "arbitrary  function."  But  Euler,  who  in  the  following  year 
took  up  the  question,  f  recognized  the  fact  that  /"could  be  perfectly  arbitrary 
and  said  that  d'Alembert  had  imposed  an  unnecessary  restriction  .J 

In  the  same  article,  as  a  special  case,  Euler  gave,  for  the  first  time,  a  solu- 
tion in  the  form  of  a  trigonometric  series.  §  He  showed  that  if  the  initial 
position  was  expressed  by 

.     tzx    ,    n   .    2ttx    ,        .    3nx    , 
y=asin—  +0sm— +  r  sin  —  +  .  .  .  , 

*Memoires  de  VAcad.  de  Berlin,  1747,  p.  214.  t  Ibid.,  1748,  p.  69. 

%Ibid.,  1748,  p.  70.  (See  also  Mimoires,  1753,  §§  III,  V,  IX,  pp.  197-200.)  In  this 
way  the  broader  conception  of  an  arbitrary  function  which  we  have  at  the  present  time 
was  first  introduced. 

§M6moire&  de  VAcad.  de  Berlin,  1748,  p.  85. 


where  a  is  the  length  of  the  chord  and  a,  /9,  y ,  .  .  .  are  any  constants,  the  form 
of  the  chord  after  a  time  t  would  be  given  by  the  equation 

.     7tx          itv    ,    n   ,     2rcx         2tcv   .        .    3jtx        Sttv   ,  ,_. 

v  =  «sm cos f-  P  Sln cos r  T  Sln cos !-•••>      (2) 

CI  CL  €L  CL  Cb  CL 

where  v  is  a  constant  multiplied  by  t.  (It  is  readily  seen  that  if  v  =  ct,  (2)  is  a 
solution  of  the  differential  equation  (1).)  This  solution  Bernoulli,  who  next 
took  up  the  subject,  claimed  was  perfectly  general,*  and  later  Lagrange,  from  a 
comparison  of  the  two  different  forms  of  solution,  was  led  to  believet  that  a 
given  analytical  function  could  always  be  represented  by  a  trigonometric  series. 

Not  much  progress  was  made  in  the  discussion  of  this  question  until  the 
time  of  Fourier,  who  in  1807  made  to  the  French  Academy  of  Sciences  the 
announcement : 

Every  arbitrary  function  of  a  variable,  whether  simple  or  composed  of  any 
number  of  different  parts  defined  by  different  laws,  can  be  represented  by  a  trigo- 
nometric series. 

The  well  known  method  by  which  Fourier  obtained  the  coefficients  an ,  bn 
of  the  trigonometric  development 

f(x)  =  X  (an  sin  nx  -f-  bn  cos  nx) 

n  =  0 

was  by  multiplying  this  equation  by  sin  nx  and  cos  nx  respectively  and  inte- 
grating from  —  rt  to  tt.%     In  this  way  is  obtained  readily 

1  [n  i   p  If"' 

60=r— -  \f[d)da;  an=. —    /(«)  sin  na  da;  bn=z —  /(«)  cos na  da.      (3) 

This  method,  however,  is  not  due  to  Fourier.  His  merit  lies  principally  in  the 
fact  that  he  was  the  first  to  recognize  the  possibility  of  representing  a  completely 
arbitrary  function  by  a  trigonometric  series. § 

Fourier  did  not  prove  generally  that  the  series  obtained  converged  to  the 
value  of  the  function.  He  showed,  however,  in  several  examples  that  this  was 
so,  and  he  considered  that  in  the  actual  application  of  the  development  to  any 
particular  case  the  proof  of  the  convergence  would  be  easy. 

The  first  important  contribution  to  the  theory  of  trigonometric  series  after 
Fourier's  work  upon  this  subject,  was  Dirichlet's  memoir,  published  in  Crelle's 

*  Memoirea  de  VAcad.  de  Berlin,  1753,  p.  157,  §  XIII. 

t Miscellanea  Taurinensia,  Vol.  Ill,  Pars  Math.,  p.  221,  Art.  XXV. 

%  Fourier  first  obtained  the  coefficients  by  integrating  from  0  to  *  {Theory  of  Heat, 
Arts.  219-221).  Afterwards  he  showed  that  they  might  be  obtained  by  integrating  from 
—  7T  to  n  (Art.  231). 

§See  in  regard  to  this  Arnold  Sachse's  memoir,  Bulletin  des  8ci.  Math.,  1880,  p.  47. 


Journal  in  1829.*  In  this  memoir  he  demonstrated  rigorously  for  the  first 
time  the  possibility  of  representing  by  trigonometric  series  functions  which,  in 
the  interval  under  consideration,  fulfil  the  three  conditions,  known  as  Dirich- 
let's  conditions,  of  being  finite,  possessing  only  a  finite  number  of  points  of  dis- 
continuity and  only  a  finite  number  of  maxima  and  minima.  His  demonstra- 
tion was  based  upon  the  proof  that  the  following  two  equations 

n*^^fto^*3P=*  [0<,<W1]  (4) 

are  satisfied  whenever  f  fulfils  the  above  conditions.  His  demonstration 
showed  that  whenever  /  satisfies  these  two  equations  the  trigonometric  develop- 
ment is  applicable.  A  few  years  later  (1837)  he  showedf  that  the  function 
might  become  infinite  for  isolated  points  between  zero  and  h  provided  that  the 

integral  ns 

/(/?)#= JF09) 

Jo 
remain  finite  and  continuous  as  /9  varies  from  zero  to  h. 

The  next  important  memoir  published  on  this  subject  was  that  of  Lip- 
schitz  which  appeared  in  Crelle's  Journal  in  1864.J  In  this  paper  he  made  an 
important  extension  to  the  class  of  functions  which  satisfy  equations  (4),  and  to 
which  therefore  Fourier's  development  can  be  applied.  He  presented  a  theorem§ 
which  was,  in  effect,  as  follows : 

If  the  function  f  (ft)  is  of  such  a  nature  that 


l/(/5)l<-4 


=  0 


g<P<h 

0<g<h< 


Tl 


tmi£i±J^mi<B  [,<*<»],  (5). 

where  A,  B ,  a  denote  finite  positive  constants,  then 

fe=oo  Jflr         sin/9    r 
equals  -~-/(0)  or  zero  according  as g>0. 

The  theorem  thus  stated  is  not  quite  accurate,  however,  since  simple  con- 
tinuity of  the  function  at  the  lower  limit  of  the  integral  is  not  sufficient,  as  is 
shown  by  the  illustration  given  on  page  21  of  this  paper.     In  fact,  Lipschitz's 

*  Vol.  IV,  p.  157 ;  Werke,  Vol.  I,  p.  117. 
t  Crelle's  Journal,  XVII,  p.  54 ;  Werke,  Vol.  I,  p.  305. 

%  Vol.  LXIII,  p.  296.  >^"\"« '*  *  *^S 

§P.  301.  jT         *'  *** 

f    I 


demonstration  requires  that  the  difference/ (0  -4-  d)  — /(0)  when  d  tends  to  zero 
approach  zero  with  a  certain  degree  of  rapidity,*  a  point  which  he  apparently 
overlooked.  This  defect,  however,  is  removed  at  once  by  making  the  condition 
(5)  include  the  limits. f  The  class  of  functions  satisfying  this  condition  includes 
many  functions  having  an  infinite  number  of  maxima  and  minima  which  would 
be  excluded  by  Dirichlet's  conditions. 

In  1867  Riemann's  well  known  work  upon  trigonometric  series  was  pub- 
lished.;}: After  an  historical  sketch  he  considered  the  question  :  What  must  be 
the  properties  of  a  function  which  is  supposed  to  be  already  represented  by 
Fourier's  series  ?  From  this  point  of  view  Riemann  arrived  at  the  important 
conclusion§  that  the  convergence  of  the  series  for  any  particular  value  of  the 
variable  depends  only  upon  the  behavior  of  the  function  in  the  vicinity  of  that 
value. 

Attention  was  now  directed  more  particularly  to  the  nature  of  the  con- 
vergence of  Fourier's  series,  and  to  the  question  as  to  whether  the  development 
was  unique.  Heine  first  demonstrated  in  1870||  that  a  function  satisfying 
Dirichlet's  conditions  possesses  a  development  uniformly  convergent  in  each 
interval  comprised  within  the  interval  ( — tt,  tt)  and  containing  no  point  of 
discontinuity  for  the  function.  He  also  showed  that  there  can  be  only  one  such 
development  by  showing  that  a  uniformly  convergent  development  representing 
zero  except  for  a  finite  number  of  points  cannot  exist,  that  each  coefficient  must 
be  identically  zero. 

Shortly  afterwards  Cantor  proved**  the  more  general  theorem  that  any 
trigonometric  series  of  the  form 

n  =co 

2  (cn  sin  ruc-f-  dn  cos  nx),  (6) 

n  =0 

*  See  the  first  two  of  his  inequalities  on  the  top  of  page  307.  For  his  proof  it  is 
necessary  that  lim  ^i^  log  2m  =  0 . 

a+u  a+ /*       1      (2rmr\o. 

fit  can  then  be  easily  shown  that  lim  — g-  log  2m  =  0.     For  — g —  <-g  B\    %    1 

and,  if  we  write  */k =  m  +  ff  [0 <  <r  <  1] ,  we  have 

t      (m\a  i  i-  logm log  m 

hmUfc)   log  *  =  lim pr+  >zma  +  ay = ^-^T  =  °  • 

\  m  > 

%  Abhandlungen  der  Getelhchaft  der  WissenscJiaft  zu  Oottingen,  1867,  Math.  Glasse,  p.  87. 
It  is  also  published  in  Riemann's  Werke,  p.  213,  and  Bulletin  des  Sci.  Math.,  1873,  Vol.  V, 
p.  20.  Although  not  published  until  1867,  it  had  been  written  for  some  time,  having  been 
presented  by  Riemann  in  1854  for  admission  to  the  Faculty  of  Philosophy  at  the  Uni- 
versity of  Gottingen. 

§  Werke,  p.  239 ;  Bulletin  des  Sci.  Math.,  1873,  Vol.  V,  p.  82. 

|  Crelle's  Journal,  LXXI,  p.  353,  §§  7,  8,  9. 

**Crelle's  Journal,  LXXII,  p.  139;  LXXIII,  p.  294. 


5  (' 


S 


convergent  and  representing  zero,  except  for  a  finite  number  of  values  of  x , 
cannot  exist.  This  theorem  Cantor  soon  extended*  still  further  to  the  case 
where  the  series  represents  zero  except  for  values  of  x  corresponding  to  the 
points  of  a  system  of  points  P  of  the  uth  speciesf  comprised  within  the  interval. 
Finally  in  1875  du  Bois-Reymond|  completed  this  part  of  the  theory  very 
satisfactorily  by  showing  that  whenever  a  function  /  can  be  represented  by  a 
series  of  the  form  (6),  the  coefficients  must  always  have  the  definite  form  (3),  thus 
showing  that  the  development  is  unique. 

Some  other  interesting  results,  which  I  will  briefly  mention,  have  been 
obtained  more  recently  in  other  directions.  In  the  Comptes  rendus  for  1881, 
p.  228,  M.  Camille  Jordan  has  shown  that  functions  having  limited  oscillation 
satisfy  Dirichlet's  equations  (4)  and  hence  are  developable  in  Fourier's  series. 
He  also  showed  by  an  example  that  the  class  of  functions  possessing  the  property 
of  limited  oscillation  includes  some  functions  having  an  infinite  number  of  dis- 
continuities scattered  all  along  over  a  finite  interval. 

In  the  same  volume  of  the  Comptes  rendus,%  du  Bois-Reymond  gives  an 
account  of  the  results  of  his  researches  upon  integrals  similar  to  Dirichlet's,  but 
more  general.  He  obtained  as  a  sufficient  condition  that  equations  similar  to 
(4)  be  satisfied, 

limfe^-mod[/(«)-/(0)]  =  0. 

e  =  0J()      a 

Du  Bois-Reymond  has  also  investigated  functions  of  the  form 

cos  <p  (x) 

pW* 

where  p  (x)  and  <p  (x)  tend  to  infinity  as  x  tends  to  zero,  />  (x)  being  always  posi- 
tive. He  has  shown  that  there  are  functions  of  this  kind  which  cannot  be 
represented  by  Fourier's  series  at  the  point  x  =  0 . 

Holder,  Kronecker,  Weierstrass  and  some  others  have  also  written  upon 
the  subject,  but  it  is  not  necessary  to  dwell  upon  their  work  here.     Kronecker || 

* Malhematische  Annalen,  V,  p.  123  ;  Acta  Mathematica,  II,  p.  336. 

t  If  P  comprises  an  infinite  number  of  points,  then  there  must  be  one  or  more  points, 
called  point  limits,  in  the  neighborhood  of  which  there  are  an  infinite  number  of  the 
points  of  P.  The  aggregate  of  these  point-limits  is  called  the  first  derived  system  of  P 
and  is  denoted  by  P.  The  first  derived  system  of  P  is  called  the  second  derived  system 
of  P  and  is  denoted  by  P",  and  so  on.  If  P^  is  the  last  derived  system  which  P  admits, 
i.  e.,  if  P^  comprises  only  a  finite  number  of  points,  P  is  said  to  be  of  the  vth-  species. 

%Beweis  class  die  Coefflcienten,  etc.,  Abhandl.  d.  k.  bayer  Akad.  d.  W.,  1875,  Vol.  XII, 
pp.  117-167. 

§  Pages  915,  962;  see  also  his  memoir,  TJntersuchungen  uber  die  Convergenz  und  Diver- 
gem  der  Fourierschen  Darstellungsformeln,  Abhandl.  d.  k.  bayer  Akad.,  Vol.  XII,  p.  1. 

||  Sitzungsberichte  der  Ak.  der  W.  zu  Berlin,  1885,  p.  641. 


6 

obtained  in  a  number  of  different  forms,  conditions  that  equations  (4)  be  satisfied, 
and  in  the  course  of  his  paper  showed  that  his  forms  included  many  of  the  con- 
ditions which  had  already  been  given  by  others. 

2.  In  considering  the  development  of  a  function  f(x)  in  trigonometric  series 
going  according  to  the  sines  and  cosines  of  increasing  multiples  of  re,  it  is 
natural  to  consider  first  what  form,  if  f{x)  is  thus  developable,  the  coefficients 
of  the  sines  and  cosines  will  take.  With  this  question  naturally  arises  another, 
viz :  Can  the  coefficients  take  more  than  one  form  ?  i.  e.  Is  the  development 
unique  ?  As  these  questions  are  answered  very  satisfactorily  by  an  important 
theorem  published  by  du  Bois-Reymond  in  1875,*  I  will  give  that  theorem  here. 

In  whatever  manner  a  function  f  (x)  can  be  developed  in  the  series 

f{x)  =  X  («»  sin  nx  -f-  bn  cos  nx) ,  (7) 

n  =  0 

holding  within  the  interval  ( — 7r,  tt)}  whose  coefficients  an)  bn  become  infinitely  smallf 
with  — ,  the  coefficients  have  always  the  form 

If"  1  r"  If" 

b0  =  ^-  I   f{a)  da ;    anz=: —  / (a)  sin  na  da ;    bn=z —  / (a)  cos  na  da , 

&X)  —  n  K  J  —  jr  7T  J—  7T 

provided  that  the  integrals  have  sense.% 

The  proof  of  this  theorem  is  as  follows: 

Form  the  function 

rpr  \       i    x2      "^T  an  sin  nx  4-  bn  cos  nx  /Q\ 

F(x)  =  bQ  —  —  2  22 J_» (8) 

Li  n  =  1  IV 

derived  from  (7)  by  two  successive  integrations.  The  function  F(x)  possesses 
the  following  properties  :§ 

1°.  It  is  uniformly  convergent  for  every  value  of  #. 

2°.    lim  F(x  +  £)  +  F&  —  £)  —  21?0c)  —  f^x)  except  for  values  of  x  which 

e  =  0  £~ 

make  the  series  (7)  divergent  or  discontinuous. 

*  Abhandl.  der  k.  bayer  Akad.  d.  W.,  Vol.  XII,  1875,  pp.  117-167,  Beweis  dass  die  Co- 
efficienten,  etc.  As  I  have  not  had  access  to  du  Bois-Reymond' s  memoir,  I  follow  here 
Sachse's  presentation  of  the  proof  {Bulletin  des  Sei.  Math.,  1880,  p.  104).  Since  this  is 
rather  condensed,  I  have  expanded  it  a  little  in  a  few  places.  In  particular,  I  have 
given  a  proof  for  the  case  where  the  function  becomes  infinite,  du  Bois-Reymond's  treat- 
ment of  which  Sachse  has  omitted. 

tThis  will  be  the  case  if  they  will  have  a  form  like  that  given  in  the  theorem,  as  is 
shown  in  the  footnote  on  page  8. 

X  This  requires  of  course  that/(a)  be  integrable  in  the  interval  ( — tt  ,  n) . 

§  Shown  in  Riemann's  Math.  Werke,  pp.  231-34  :  Bull,  des  Sci.  Math.,  Vol.  V,  1873,  pp. 
41-45  ;  Picard,  Traite  d'  Analyse,  Vol.  I,  pp.  240-44. 


7 
3°.    lim  F{x  +  e)+F(z-*)-2F(x)  =  Q  for  yalue  of  ^ 

e  =  0  £ 

Let  us  consider  first  a  function  f(x)  which  is  finite  and  continuous  and 
does  not  have  an  infinite  number  of  maxima  and  minima.  We  must  have 
necessarily  for  every  value  of  x  between  —  re  and  -f-  rt , 

^\Xdafe(P)dp==f(x).  (9) 

Form  the  expression 

0{x)  =  F(x)—  Pda  [/(/?)  d/3. 

J IT     *  —  It 

From  the  property  2°  and  from  (9)  it  follows  that 

lim  0(*  +  e)-20(*)+0(*-g)=lim  4*0  {x)  _  Q 
«  =  o  e2  e=o      s2 

and  therefore 

0(x)=ZCo-\-C1X. 

Hence 

F{x)  =  [*d«  f>(/9)  dp  +  c0  +  ^r.  (10) 

»  —  ir   «  —  it 

If  now,  according  to  the  method  employed  by  Fourier,  we  multiply  (8)  by 
sin  nx  and  cos  nx  successively,  and  integrate  from  —  rz  to  -\-tt  ,we  will  get 

IF*1 


flT  .  yjS  Ctt 

F(a)  da  =:  -K-  &o>        -^(°0  s^11  wa  ^a  = 

J  — It  "  J_„- 

J>(«)  cos  n«  da  =  -^f +(-!)» 


In  these  equations  substitute  for  F(a)  its  value  given  in  (10)  and  integrate 
by  parts. 

[V  (a;)  <&>  =  [*     {^da[f{P)dp+Co+<w\dx',    f  "da  fda  f/(/9)dj8 

J T  J — JT     V   J IT     J IT  J  J  — IT     J  —  It      J  —  It 

=  x  [da  |>(/3) d/9l-  Hwte  [>(/?) d^ 

J  —  it   J  —  7r  — ' —  it    J  —  it       J  —  it 

=  ^j>^)dp]^-^|V(a)d«-^J>(/9)d/9]^+  \\^f{x)  dx 
=  ~2){71  —  aff (a) da >    I (°o  +  G&) dx  =  2ttc0 . 

\[{n-aff{a)da  +  2r:c,=  ^-b,.  (12) 


Hence 


*  Integrating  by  parts,  it  is  readily  seen  that 

[n     x*  2k        f      xH 

-5-  cos  na;  da:  =  (—  l)n  — r  ,  —  sin  na;  da;  =  0 , 

J-1T    Z  n  J  —  *       2 


j  F(a)  sin  na  da  =  I  sin  na  d«  \     da   /(£)  dp  -f-  c0  +  cy/  > 

=  {  i-  cos  na  [|  d«  J>  (/?)  ^  +  c0  +  *«]  }  * 

+  -i-j  ow  na  [  J/(0)  d£  +  Cl  J  da 
=  (:=^T1&  fc»^+  (~1)nr27rCl  -^f/Wrinnaci, 

"  J— IT  -1— 7T  ",  J—  IT 

T" r   /  (a)  sin  wa  da . 


Hence 


Therefore 

(_^n  +  l  ^  |y(^  ^  _  |*^w  ^  +  2^-|  _  _^^  ^  nada  =  __W.   (13) 

.F(a)  cos  nadaz=     cos  na  da  *!     da  1/ (/3)  d,3  -f~  Co  -f-  cyz  > 
=  {  ~n~  Sin  naU^"  \l^  ^  +  co  +  ci«]  }  _ 

—  {  W  C°S  na  L J/f  ^  ^  +  Cl]  }  _  —  ^2-j  /(«)  C08  W«  d«' 

^  j/ja)  da  -  -w\ha)  cos  wa  da = -  hf + (- i)n  IF  •   (14) 

From  the  three  equations  (12),  (13),  (14)  we  can  readily  obtain  the  values 
of  c0,  Ci,  60,  an,bn,  by  remembering  that  these  equations  are  true  for  all  integer 
values  of  n  greater  than  zero,  and  that  an)  bn  and  the  integrals 

|V  fir 

/(a)  sin  na  da,     f(a)  cos  na  da 

J — It  J — IT 

tend  to  zero  with *     Thus  from  equation  (14)  by  making  n  tend  to  infinity 

*That  these  integrals  tend  to  zero  whenever/(a:)  is  integrable  is  proved  by  the  same 

method  as  the  theorem  given  on  page  14.    In  the  integral     f(a)  sin  na  da,  f{a)  takes 

the  place  of  the  function     •        in  the  proof   of   that   theorem.      For   the   integral 

/(«)  cos  nada  it  is  only  necessary  in  that  proof  to  make  the  subdivision  of  the  interval 
J-*  7T 

( — 7T,  »r)  in  such  a  way  that  each  small  interval  equals  #~and  the  points  of  subdivision 
are  integer  multiples  of  -g  . 


we  see  first  that 


b°=itr\ha)da' 


and  that  then  we  obtain  the  value  of  bn  for  any  value  of  n.     The  results  which 
we  obtain  from  the  three  equations  are 

fc=4^j/ («)["§ (7r  —  af~\  da>    *=j£/W(*— *)«ki 

If"  i  pf,  if"' 

60=: -^-^/(ajda,     aM  =  — I  /'(«)  sin  nada,       bn  =  — \  f  (a)  cos  na  da . 

If  now,  however,  supposing  that  f  (x)  is  finite,  we  subject  it  to  the  single 
additional  condition  of  being  integrable,  the  above  proof  that  (P  (x)  =  c0-\-  cxx 
will  not  hold.  We  must  make  another  investigation  for  this  case.  f{x)  will 
not  have  necessarily  at  each  point  a  determinate  limit,  but  its  value  will  lie 
between  a  superior  and  an  inferior  limit.  Let  us  designate  the  half-sum  of 
these  two  limits  by  S  (x) ,  the  half-difference  by  D  (x).  We  can  evidently  give 
to  the  function  the  form  S  (x)  -f  jD  (x) ,  where  j  denotes  a  real  number  lying 
between  —  1  and  -j-  1 .     Let  us  now,  according  to  the  method  employed  by 

Biemann,  find  an  expression  for  lim  — -^-^  .     Put 


e  =  0 


X  («n  sin  nx  -f-  bn  cos  nx)  —  S  (x)  -j-  a\  • 
»=o 

Taking  3  an  arbitrary  small  quantity,  we  can  fiud  a  value  of  m  such  that 
|  am  |  <  D  (x)  -|-  d .     We  can  now  write* 


J2F{x)       -,., 
.2      =/»  + 


sin  (n  —  1)  -( 


sin  n  -jr- 


2^ 


Taking  e  sufficiently  small  to  have  m  -^  <[  tt  ,  let  us  divide  this  series  into 

three  parts.     In  the  first  let  n  increase  from  1  torn;  in  the  second  from  m  -\-  1 

to  s,  the  greatest  integer  in  -= — ,  and  in  the  third  from  s  -j-  1  to  infinity.     The 

first  part  will  tend  to  zero  as  e  diminishes.     The  second  part  will  be  less  in 
absolute  value  than 


*  Riemann,  Math.  Werke,   p.   223 ;  Bui.  des 
d' Analyse,  Vol.  I,  p.  241. 


4.  Math.,  1873,  p.  43;  Picard,  Traiti 


10 
The  third  part  will  be  less  in  absolute  value  than 

Now  passing  to  the  limit  we  see  that  the  first  of  these  expressions  will  tend  to 
D  (x)  -f  8 ,  and  the  second  to  (*  +  — )  (D  (x)  -f  8) .     Hence 

ta^^^/sf(«)+i(i+-y  +  -i-)p>W+fl      [— i<i<i]. 

Putting 

F1(x)z=\Xda\f^)d^ 

•  —  IT    J  —  It 

we  have  that 

lim  -^-'  =  S(z)  +jxD  (x)        [- 1  <j,  <  1] . 

But  0  (x)  =  F  (x)  —  Fx  (x) ,  and  hence,  neglecting  the  arbitrary  small  quantity 
8 ,  the  modulus  of  the  greatest  value  which  lim  ^  can  take  is 

e  =  0  £ 


(a  +  ^r  +  T>W- 


We  can  now,  by  making  use  of  the  condition  that/" (x)  is  integrable,  show 
that  <P  (x)  is  a  linear  function  of  x .  Let  us  divide  the  interval  (x ,  x  -f-  a)  into 
smaller  intervals  limited  by  the  points  x ,  #  -f-  ^ ,  x  +  ^i  +  <^  >  •  •  •  >  #  "h  ^i 
-{-...  -f<5ft_i ,  a;  -}-  a  and  form  the  sum 

dlD(z+PldJ  +  dtD{z  +  3l  +  pA)  +  ...  +  8J)(x  +  8i  +  l+.--+pM,  (15) 

where  the  quantities  p  are  positive  fractions.  This  sum,  in  virtue  of  the  con- 
dition of  integrability,  must  tend  to  zero  with  8.  Hence  when  the  <5's  become 
infinitely  small  and  e  tends  to  zero,  the  limit  of 

J>[d10(x  +  p1d1)  +  d90{x  +  31  +  pA)+>  •  .+8n4>{x  +  81  +  •  -  -+ftA)] 

?  ' 

which  in  absolute  value  cannot  exceed  the  sum  (15)  multiplied  by  f  2  -f-  —$-  -\ J, 

must  be  zero.     That  is,  we  must  have 

a0(x  +  a)da  =  Q. 


lim  — y 

e  =  0    £ 


It  follows  therefore  that 

\  0  (x -\- a)  da=  c„ -\- CxX .  ^V 

Jo 


11  / 

But  since  F(x)  and  Fx  (x)  are  continuous  functions  of  a;  ,*  <P{x),  =  F(x%£~Fx  (x) 
is  continuous  also.     Hence,  employing  the  theorem  of  means,  we  can  write 


<P(x  +  a)da  =  a0(x  +  aJ         [0<«1<a]. 
Jo 
Consequently 

Now  let  a  tend  to  zero.     The  first  member  will  tend  to  a  definite  limit,  and 

therefore  the  second  member  will  also,  •'.  e.,  —  and  —   will  tend  to  fixed  quan- 

a  a 

tities  c0'  and  c/'.     We  have  then 

4>(x)  =  c0'  +  Gl'x, 

and  the  demonstration  can  be  completed  as  before. 

Suppose  now  that /"(a)  becomes  infinite  in  the  interval  ( — tz  ,  tz) ,  but  still 
satisfies  the  condition  of  integrability.  Over  each  portion  of  the  interval  con- 
taining no  infinite  point  fovf{x)  0  (x)  is  a  linear  function  of  x  by  the  above 
proof.  Hence  since  (P  (x)  is  continuous  in  the  interval  ( —  tz ,  tz),  the  curve 
y=0(x),&sx  varies  from  —  tz  to  tz  ,  if  it  does  not  represent  a  straight  line,  at 
least  represents  a  continuous  broken  line,  the  corners  corresponding  to  values 
of  x  which  makey(.r)  infinite.    We  will  show  that  the  line  is  straight. 


Writing  \"A?W=  rW, 


it  is  easy  to  show  that  ?F  (a)  is  continuous  in  the  interval  ( — tz  ,  tz)  .     For 

V(a  +  d)-¥{a)  =  ^f{P)dp 

can  evidently  be  made  less  than  any  assigned  small  quantity  s  by  choosiug  d 
sufficiently  small  since,  if,/  (x)  becomes  infinite  for  x=zx0,  the  condition  of 
integrability  requires  that 

lim  [/•(/?)  d^  =  0,    lim  f/t/3)  W = 0  ■ 

Employing  the  theorem  of  means,  and  remembering  that    $  '(a)  is  con- 
tinuous, we  get  at  once 

lim  -ife)  —  lim  -~|V  (a)  da  =z  lim  i-[  [*¥*(  a)  da  —  fV  (a)  da]  =  0 . 


*Fi  {x)  is  continuous  whenever  /  is  integrable,  since  JFVis  the  integral  from  — -  to  x 
of  a  finite  function. 


12 
But  since  by  property  3°,  p.  7,  lim ^ '  =  0 ,  we  must  have 

e  =  0  * 

That  is,  Hm  0(x  +  *)—0(x)  =  Hm  0(x)-0(x-e) 

which  tells  us  that  the  directions  of  the  two  straight  lines  meeting  at  any  vertex 
are  the  same.  Hence,  for  values  of  x  in  the  interval  ( —  tt  ,  tt)  ,  y  z=z  0  (x)  repre- 
sents a  single  straight  line,  and  we  can  write 

0  (x)  =  c  +  dx 
and  complete  the  demonstration  as  in  the  first  case. 

The  theorem  just  proved  shows  that  whenever /"(a;)  is  integrable  and  can 
be  represented  by  a  development  of  the  form  (7),  the  coefficients  an  and  bn  in 
this  development  must  always  be  of  the  form  originally  given  by  Fourier.  But 
the  integrals  giving  these  coefficients  are  perfectly  determinate  quantities.  It 
follows  therefore  that  a  given  function  can  be  developed  in  a  trigonometric 
series  of  the  nature  mentioned  in  only  one  way,  or,  in  other  words,  that  the 
development  (7)  must  be  unique. 

We  have  considered  above  a  development  available  in  the  interval 
( —  re  t  1?) .  If,  however,  we  had  desired  to  obtain  the  form  of  the  coefficients  in 
a  development  holding  for  some  other  interval  of  2~ ,  say  [m- ,  (m  -f-  2)  ;:] 
where  m  is  an  integer,  the  method  of  obtaining  them  would  have  been  precisely 
the  same  except  that  instead  of  integrating  over  the  interval  ( — -,  -),  as  on 
page  7,  we  would  have  integrated  over  the  interval  [mr: ,  (m  -|-  2)  tt]  .  If  we 
should  do  this,  getting  first  equations  similar  to  (11),  and  then  after  partial 
integration  equations  similar  to  (12),  (13)  and  (14),  we  would  find  that  while 
the  expressions  for  c0  and  c,  would  be  different,  60,  an  and  bn  would  be  the  same 
as  before  (p.  9),  except  that  the  field  of  integration  would  be  [m~ ,  (m  -j-  2)  ;r]  .* 

3.  We  have  seen  that  if  a  function  f  (x)  is  capable  of  being  represented  by 
a  trigonometric  series  of  the  form  (7),  this  development  must  be 


f(x)=^f(a)da 


sin  x  \f{a)  sin  a  da-\-  sin  2x  \f{a)  sin  2a  da  -\-.  .  . 
+  —  {  J-'r  }~n  }  .       (16) 

i  +  cos  x  \f{a)  cos  a  da-\-  cos  2x  !/"(«)  cos  2ada-\-  .  .  .  J 

♦Evidently  also  a  development  of/ (x)t  available  in  the  neighborhood  of  any  point 
a ,  is  at  once  obtained  by  developing  f(a  +  x) ,  regarded  as  a  new  function  of  x  ,  in  the 
ordinary  manner.  The  two  developments,  holding  for  the  interval  [rmr ,  (r»  +  2)  t]  , 
obtained  by  the  two  methods  are  easily  shown  to  be  identical. 


13 


We  need  then  to  consider  under  what  conditions  this  series  will  converge  to  the 
value  of  the  function  f(x).     The  sum  of  the  first  n  terms,  Sn  say,  gives  us 

1  F 

8n=z  —   [|  +  cos  («  —  x)  -f  cos  2  («  —  x)  -\-  .  .  .  -f-  cos  n (a  —  x)\f  (a)  da 

"■    J  —  ir 


*  f(a)  sin  (2ro  +  1)  %-?  da 


2  sin 


a  —  x 


Let  us  find  to  what  limit  8n  tends  when  n  increases  indefinitely.     By  making 
a  change  of  variable  Sn  can  be  written 

s  -   1  p*<— *>/(a?  +  2r)  sin  (2n  +1)  7  <fr 


+ 


J_f*('-*)/(a;  +  2/')  sin(2n+l)rc?r 

7T  J0  sin  p 

jj^  rt  (*  +  *)f(x  _  2r)  sin  (2n  +  1 )  r  <*r 

7T  Jo  sin  t' 


(17) 


where  —  tt<^x<^k  .  This  form  of  expressing  Sn  leads  one  very  naturally 
to  consider  the  limit  for  k  =.  oo  of  the  following  integral,  known  as  Dirichlet's 
integral, 

(>?(r)smkrdr      [0<i<,T], 


I" 

JO 


;o      sin  y 

where  <p  (y)  has  been  put  for/" (x  ±  2y) .     We  will  consider  under  what  circum- 
stances 

(18) 


Hm  pyQQBinMr  _  lim  JL  .  M . 
fc=ocJo         sin  y  e=r0   2 


For  if  (18)  holds  for  all  values  of  h  lying  between  0  and  iz ,  then  when  n 
increases  indefinitely,  Sn  will  tend  to  the  limit 

lim  *[/(*  +  &)  +/(*-  2*)], 

6  =  0 

i.  e.,  to  y(»)  unless  /*  has  a  point  of  discontinuity  at  the  point  x,  and  in  this 
case  to  a  mean  between  the  two  values  taken  by  f  at  the  point. 

It  is  only  necessary,  however,  to  consider  0</t<  -^- ,  for  if  (18)  holds  for 

such  values  of  h,  it  will  hold  also  when  ~  <^h<^7r.     In  order  to  see  this, 

suppose  (18)  is  true  when  0  <  h  <  -&  •     This  will  evidently  require  that 


14 
If  now  -jj-  <  h  <  z ,  then 

f*  eg  (r)  sin  &r  ^  __  f »  <p  (y)  sin  fr  d<<   ,   (h<p(r)  sin  kr  d 
Jo       sin?-         '       Jo       sin-/-         '       J,      sin  y 

In  the  case  where  h  is  an  odd  integer,  h  =  2n  -{-  1 ,  the  second  integral  in  the 
right-hand  member  of  this  last  equation  will  be  zero,  for,  on  changing  y  into 
~  — y,  it  will  become 

ir 

p     <p  (tt  —  y)  sin  ky  dy 
iv-h  sin?- 

which  is  zero  by  (19),  provided  that,  over  the  interval,  tp  (tc —  y)  fulfil  the  con- 
ditions required  of  <p  by  equation  (19). 

4.  We  have  shown  in  the  last  section  that  the  question  of  the  convergence 
of  the  trigonometric  series  (16)  to  the  value  f{x)  reduces  to  the  consideration  of 
the  circumstances  under  which 

Km  py(r)*»*r<fr     lim  *    ()      r0  <  h    «n .  (20) 

&=Jo         sinr  e=0  2  Yyj         L   ^         2  J 

We  will  now  show  that  it  is  only  necessary  to  consider  what  conditions  <p  must 
fulfil  in  order  that  (20)  be  satisfied  when  A  is  a  quantity  greater  than  zero,  but 
as  small  as  we  please.     This  can  be  seen  at  once  from  the  following  theorem  : 
Whenever  <p  (y)  is  integrable  over  the  interval  (e,  h)  and  (p  (s)  is  finite, 

lim[V(r) sin  hdy  =  0        r0<e<A<  *n 

k=Je         Biny  L    ^    ^  2  J 

To  show  this  let  us  divide  the  interval  (e ,  h)  into  n  parts  d1}  d2)  .  .  .  dn, 

making  the  points  of  division  odd  integer  multiples  of  --^j- ,  distant  from  each 

other  by  an  amount  equal  to  --.-   .     Let  us  also  take  s  itself  an  odd  integer  mul- 

tiple  of-KT-,  so  that  we  shall  have 

flj  =  o2  ==  ds  =  .  .  .  =  on_!  =  -j-  >  3n . 

We  will  now  seek  the  limit  of 

(h  <p(y)  sin  ley  dy  ^ 

}(         smy 

as  k  tends  to  infinity.  Let  yp  denote  the  value  of  y,  in  the  interval  3P)  for 
which  <p  (y)  approaches  nearest  to  zero,  and  in  each  interval  dp  let 


sin 


(r)_  y(rP),  jrf(r)i 

u  y       sin  yp    '       Lsin  y]  ' 


15 
We  can  evidently  express  the  integral  (21)  as  the  following  sum  of  integrals, 

Tt^£**]+f  *'i^**+f  >*£?*■  (22) 

where  ^  -~  =  £ ,  )H + x  =  J,  +  2 ,  /„  -^  =  A  —  on .     But  since,  in  virtue  of  the 


condition  of  integrability, 


<P  Op)  $p 


**** 


the  first  term  of  (22)  is  zero  because 


8111^^  =  —  np"  J  =»• 


Now  let  fc  increase  indefinitely.     The  first  term  of  (22)  will  continue  to  be  zero. 
As  for  the  second  term  we  have  always 


4my^dH^m^Ad<'n&^ 


u 

Lsm  yj  '     '  |    \|e  Lsm  y. 

which  tends  to  zero  from  the  condition  of  integrability.  Finally,  for  the  last 
term  of  (22),  since  S1.n  J  varies  always  in  the  same  sense  from  h  —  dn  to  h,  we 

have,  from  Bonnet's  theorem,  that 

[h       P(r)sinfer  ,, 
]h  —  in      sin  7- 

is  not  greater  in  absolute  value  than  one  of  the  two  quantities 

sin  (A- K) fa* dr>  «*(k-aJ\?V>*r      [h-3.<e<h}. 

But  each  of  these  approaches  zero  as  h  increases  indefinitely,  even  though  <p  (y) 
becomes  infinite,  because  if  y0  is  a  point  at  which  <p  (y)  becomes  infinite,  the  con- 
dition of  integrability  requires  that 

fYo  f  Vo  +  « 

lim    <p  (y)  dy  =  0 ,    lim     <p  (y)  dy  =z  0 . 

a  =  0-lyo  —  a  n  =  0Jyo 

Hence  the  sum  (22)  tends  to  zero  as  k  becomes  infinite,  and  we  have 


lim 


'h<f{y)sinkydy_0 
sin  y 


*  *r 


[    U  ITV 


16 

This  conclusion  is  very  important,  for  (see  equation  (17),  p.  13)  it  shows 
that  if  a  function  f{x)  is  to  be  developed  in  Fourier's  series,  it  only  needs 
to  satisfy  the  condition  of  integrability  except  for  a  region  as  small  as  we 
please  on  each  side  of  the  point  at  which  the  development  is  to  be  made,  or  in 
other  words,  that  the  convergence  of  the  series  for  any  particular  value  of  x 
depends  only  upon  the  behavior  of  the  function  in  the  vicinity  of  that  value. 

A  particular  case  of  the  above  theorem  is  when  <p  (y)  is  a  constant,  say 
unity.     The  theorem  will  then  become 

lim  \hs™krdr  =  0  r0<s<><4 

*  =  «,  Je      sin  y  L  2 

This  can  also  be  shown  very  readily  by  partial  integration.* 


(23) 


5.  The  theorem  proved  in  the  preceding  section  reduces  our  problem  to  the 
consideration  of  the  conditions  which  <p  (j)  must  satisfy  in  order  that 

Hm  f-  ?(r)sWr  =  ]im  n     ((J)  =  n     {  say>  m 

k  =« Jo  sin  7*  <r=:0  ^  z 

where  £  denotes  a  quantity  as  small  as  we  please  but  greater  than  zero. 
Now,  k  being  an  odd  integer, 


sin  y 
and  hence  from  (23)  we  must  have 


7. 

P*"  sin  ky  ,  n     . 

Jo  isry^T^ 


lim  p™fedr=   7 

Jo  smr 


fc=ooJo  sin  y  2 

Accordingly  we  may  write 

lim  P  y  fr)  sin  kr  dr  -  lim  ['  <>  fr)  ~  ? ( +  °)]  Sin  hr  dr  +  g  r  (+  0) 
*=oj0         sin?'  fc=»J0  sin  y  2 

Equation  (24)  will  therefore  be  satisfied  if 

lim  [e  l>  (r)  -  <P  (+  0)1  sin  kydy_n  (26) 

jfc=ooj0  sin  y 


*  Ch  sin  ky  JL_  coaky~\h l_  f* 

Je  sin  y   dy-~    k   sin>Je        k  ]t 


1   f  a  cos  ky  cos  y  dy 


sin"  y 


which  tends  to  zero  with  -j— 


f  This  follows  at  once  from  the  equation 

sin(2?t  +  l)a;:_1  +  2cos2a!+2co84a;+       _  +  2cos2tw; 


sin  x 


Let  us  first  suppose  that  <p  is  a  function  possessing  the  property  of  limited 
oscillation  or  limited  variation,  as  Jordan  has  defined  it,*  in  the  interval  (0,  e). 
It  can  then  be  written  in  the  form 

<p(y)  =  <p(+0)  +  Py-Ny, 

where  Py  represents  the  sum  of  the  positive  oscillations  of  <p  from  zero  to  y , 
and  Ny  represents  the  sum  of  the  negative  oscillations.  In  this  case  the  left 
hand  member  of  (25)  will  take  the  form 

limHPv-^)sini^r, 
fc=ooj0  sin  y 

But  from  Bonnet's  theorem,  since  Py  is  a  positive  function,  never  decreasing, 

rfT8JnMr  =  ftW^      [0<*<«]. 

Jo        smy  ]t  sin  y  , 


Suppose  now  that  we  divide  the  interval  ( 0  ,  ~  )  into  the  partial  intervals 

(0,~f)'  \T"  "*/'  \T'  ~V)''''\T'  ¥)' 

r  being  the  largest  number  of  times  that  ~  is  contained  in  -J-  .    We  can  write 

k  2 


7i         [z  sin  ky  j  ,  ,      ,,._ 


where 


^       sin  ky  dy  \      \n  ^2        1  r         -,     „    0  n 

W)-  sm  IT 


This  gives  at  once 


Px< 


sm 


^   2  .  -     .     2 


(26) 


and  hence  whatever  value  $  takes  in  the  interval  ( 0 


0<  ?*■£*<"+-* 

J0  sm  j'  2  - 


But  since 

lim 

k  =  oo 

we  have 

lim 

k  =  00 

limpsinMr  =  ^ 

fc=ooJo    smr  2 


r 

6  sin  &;'  cZ?- 
*     sinr 


<- 


(2' 


*See  Comptes  rendus,  1881,  p.  228;  also  Jordan's  Cow«  d' Analyse,  Vol.  II,  p.  216. 


18 


It  follows  therefore  that 


Similarly 


Hence 


lim 

fc  =  oo 


lim 

k  =  oo 

lim 


£ 


Py  sin  ky 
siny 


dy 


tNy 8in  krdr 


sin  r 


< 


<\*.. 


r 

JO 


{Py 


-Ny )  sin  ky  dy 
sin  y 


<r:M, 


where  M  is  the  largest  of  the  quantities  P(  and  N€ .  It  follows  therefore  that 
(24)  will  be  satisfied  provided  that  P£  and  Ne  tend  to  zero  with  e.  This  will 
evidently  be  the  case  under  the  following  circumstances : 

1°.  If  the  function  is  continuous  and  has  only  a  finite  number  of  maxima 
and  minima. 

2°.  If  the  function  has  a  finite  number  of  discontinuities,*  but  is  finite, 
and  has  only  a  finite  number  of  maxima  and  minima. 

These  conditions  are  precisely  those  of  Dirichlet,  and  hence  we  see  that  for 
functions  satisfying  Dirichlet's  conditions  we  must  have 

limf6^)sm^r_jr      (,  Q)> 
*  =  «>  Jo         sin  y  2 

6.  If  the  function  <p  is  continuous  or  has  a  finite  number  of  discontinuities, 
but  has  an  infinite  number  of  maxima  and  minima,  it  may  still  be  true,  when  f 
possesses  the  property  of  limited  oscillation  in  the  interval  (0,  e),  that  Pe  and 
Ne  tend  to  zero  with  e.  But  for  the  general  case  of  a  function  with  an  infinite 
number  of  maxima  and  minima  (with  no  discontinuities  or  a  finite  number  of 
them),  and  also  for  the  case  of  an  infinite  number  of  discontinuities,  a  different 
investigation  is  necessary.  We  can,  however,  derive  a  very  general  condition 
which  it  is  sufficient  for  functions  of  this  kind  to  fulfil  in  order  that  the 
equation  just  written  be  satisfied. 

In  equation  (25)  we  can  replace  sin  y  by  y  since  their  ratio  is  very  near 
unity  when  e  is  very  small.     Now 


Wr) 


^(-j-0)]  sin  kydy 
T 


Jo 


f(r)-r(+o)sinkr 

r  <! 


dy 


y(r)  —  y(+Q) 
r 


dy. 


*  For  if  the  function  has  a  finite  number  of  discontinuities,  we  can  take  e  so  small 
that  in  the  interval  (0,  c)  the  function  will  be  continuous. 


19 
But  e  can  be  taken  as  small  as  we  please,  and  hence  a  sufficient  condition  that 

i8  iimf  I  y(r)— y(+Q)i  dr  — 0,*  (28) 

Let  us  compare  this  condition  with  one  or  two  other  conditions  which 
have  been  given.     Lipschitz's  condition  was 

l{mf±±^ZllSll<B,  (29) 

8  =  0  ° 

where  B  and  a  denote  any  positive  finite  quantities.     His  condition  is  less 
general  than  (28).     For  if  (29)  is  satisfied  we  can  write 


lim  r-  |pfr)-y(+o)|  rfr<Hm  f-Brdr 

«=oJo  «  =  oJo       X 

V  «  Jo 


:0J0  «  =  0^0 

But 


which  tends  to  zero  with  e.  Hence  (28)  is  satisfied  in  this  case.  But  on  the 
other  hand  (28)  can  be  satisfied  without  Lipschitz's  condition  being  fulfilled. 
Thus  if  <p  was  of  such  a  nature  that 

I!fo(log  dl  \?b  +  *)-H>  (r) ] == X, 

where  Kis  a  finite  constant,  Lipschitz's  condition  would  not  be  fulfilled,  since 

M  K 

a=o<*tt(logd)a 
The  condition  (28),  however,  would  be  satisfied  in  this  case,  for  we  can  write 

limf-k(r)-?(+0)l  d      UmKi-     dr     =  Bmr_  *_T  =  0. 

«=oJo  r  e=o    Jor(logr)      .=<>l    logr-io 

Another  condition  which  has  been  given  is,  denoting  by  D  the  difference 
1P(r  +  *)  — ?(r),  "  lim  2>log*  =  0.  (30) 

8  =  0 

The  two  conditions  (28)  and  (30)  differ  from  each  other  in  comprehensiveness 
very  little,  but  the  latter  is  slightly  more  general.  Thus  both  can  easily  be 
shown  to  be  satisfied  by  a  function  for  which 

lim  D  (log  dy—K        [«>1], 

8  =  0 


*See  page  5.  This  condition,  for  integrals  similar  to  Dirichlet's  but  more  general, 
was  obtained  by  du  Bois-Reymond  in  another  way.  See  his  article,  Comptes  rendus,  1881, 
p.  915. 


20 


where  K  is  a  finite  constant.     But  both  cease  to  be  satisfied  for  a  <  1 .     If, 
however,  we  consider  a  function  for  which 

lim  D  log  d  [log  (  —  log  <*)]•  = K        [0<a<l], 
«=o 

the  condition  (30)  will  be  satisfied  while  (28)  will  not.     For 

K 


But 


lim  D  log  d  =  lim  p^- 

5=o         &  5=o[log(  —  log^)> 


=  0 


iim  f <k _  Iim  r  ^[log(-iogr)] 

e =o Jo  r  log  r  [^g  (—  log  r)]a      £=0  Jo  [log  (—  log  r)Y 

~lim  \r~ -^[log(—  logr)]1_a  ]"=<». 

e  =  o  Li  —  a  J0 

Let  us  now  apply  the  condition  (28)  to  one  or  two  functions  having  an 
infinite   number  of  maxima  and  minima  in  a  finite  region.     The    function 

f{x)  =  x  sin  —  becomes  zero  for  x  =  0,  having  at  that  point  an  infinite  num- 
ber of  maxima  and  minima.  For  this  function  the  left-hand  member  of  (28) 
would  become,  since  for  #  =  0  <p  becomes /"(i  2f)-=.f{2y), 


But 


limf 

€=0jl 

iosiniH<E 


2sin-±-c7r. 
2r 


sin 


2r 


dT<e. 


Hence,  since  the  condition  (28)  is  satisfied,  f(x)  is  capable  of  being  represented 
by  Fourier's  series  at  the  point  zero  as  well  as  elsewhere. 
Again  consider  the  function 


/(*)  = 


(log 
For  this  case  we  have  to  consider 


x 
JtF 


y 


sin 


4  [*>!]• 


sin 


2r 


lim  i  -, J  .  „ 


r 


Now 


sin 


2T      dy 


■(*£)' r 


< 


sin 


2r 


( 


gl0°r 


dr< 


0Q- 

1  — a 


21 

which  tends  to  zero  with  s  if  a  y>  1 .  Hence,/ (a;)  is  developable  in  Fourier's 
series  at  the  point  zero. 

7.  Although  the  condition 

imposes  very  little  restriction  upon  the  function  <p ,  yet,  as  has  already  been 
mentioned,*  du  Bois-Reymond  has  shown  that  there  are  continuous  functions 
which  cannot  be  represented  by  Fourier's  series  at  every  point.  The  following 
illustration  given  by  Schwartzf  is  a  special  case  of  functions  of  this  kind  dis- 
cussed by  du  Bois-Reymond. J 

Recalling  the  value  of  the  sum  of  the  first  n  terms  of  Fourier's  series, 

8.=  U"--/(«+2r)'m h dr+  J_f">  +  y  (*-2r) sinkr  d      [i=2n+1]> 
Tt  Jo  sinr  7T  Jo  sin?-  L  J' 

and  the  theorem  given  on  page  14,  it  is  evident  that  we  only  need  to  find  an 
integrable  function,/  such  that  for  some  value  of  x, 

lim  \<f(x  +  2r)smkrdr=:  ^ 

*=°°Jo  r 

where  e  is  a  small  quantity  greater  than  zero,  and  of  such  a  nature  over  the 
interval  (x —  e,  x)  that  the  second  integral  of  Sn  is  finite. 

Divide  the  interval  f  —  ,  OJ  into  intervals  becoming  smaller  and  smaller, 

as  follows : 

IT'  (I)]'  L(T)'  (|]'-,1(T^'(|"]',-'[(^^'(^)J,L(^'0]'   (31) 

where  (X)  =  1 .  3  . 5  .  .  .  [21  -\-  1] ,  and  consider  a  function  which  in  the  Xth 
interval  is  defined  by  the  formula 

/(/3)=c,sin(;)A, 

where  the  constants  c1}  c3,  .  .  .  <v+1  are  positive  and  decrease  indefinitely  to 
zero  as  ju  becomes  infinite.  When  //=  oo  this  function,  evidently  continuous, 
tends  to  zero  with  /3,  presenting  an  infinite  number  of  maxima  and  minima  in 

*  Page  5. 

\  Bulletin  des  Sci.  Math.,  1880,  p.  109.  I  give  here  the  function  given  by  Schwartz, 
but  prove  the  divergence  of  the  series  differently. 

\  Untersuchungen  iiber  die  Convergenz  und  Divergenz  der  Fourierschen  Darstellunga- 
formeln.    Abhandl.  d.  k.  bayer  Akad.  d.  W.,  II  CI.,  XII  Bd.,  II  Abth.,  §§  35-37. 


22 

the  region  of  the  point  zero.  We  will  assume  that  for  negative  values  of 
fit  f  (ft)  ^8  °f  sucn  a  nature  that  it  fulfils  the  condition  (28).  This  is  allowable 
since  f  is  an  arbitrary  function. 

In  order  to  show  that  the  Fourier  series,  which  for  all  other  values  of  ft 
represents  the  function,  diverges  for  ,3  =  0,  we  need  to  examine  the  integral 

f'/(2r)smtr  d 
Jo        r 

We  will  show  that  this  integral,  with  proper  choice  of  the  c's,  becomes  infinite 
with  h.  To  do  this  we  will  give  to  h  only  values  of  the  form  (/j.)  ,  where  fx 
increases  indefinitely.*  Let  us  divide  the  integral  into  partial  integrals  cor- 
responding to  the  intervals  of  (31),  beginning  with  the  right,  and  let  /x  be  the 

greatest  integer  for  which  ~  >  e  .     Putting 

(xi) 

J=  nkf{2y)  sin  (fi)ydy  ^ 

Jo  T 

we  have 

J  =  c>x  +  in^sin(//  +  l)rsin(//)rdr+Al|1t7A> 

Jo  T  m 

where 

t  _     f(A~ 1)  s*n  M  f  s*n  (aO  y  dy 


w 


Over  the  interval     0  ,  j—.\  sin  (fi)  y  is  always  positive,  and  hence  applying  the 
theorem  of  means,  we  have 

it        I  _     „        (V>sin(/<+l)r  sin  {fj)rdr       -,        f^sin  {^)ydy 
Jo  Jo 

this  last  integral  by  equa 
lim  c^  +  i  =  0,  we  must  have 


TZ  2 

But  this  last  integral  by  equation  (27),  p.  17,  is  less  than  -^-  -| ,  and  since 


lim  J^  +  i^O. 

ft=O0 


•This  of  course  is  allowable,  because  increasing  k  from  («)  to  (,«  +  1)  amounts  simply 
to  adding  a  large  number  of  the  terms  of  the  series  at  one  time  instead  of  taking  only 
one  term  additional.  Thus  putting  a  equal  to  the  number  of  terms  of  the  series  taken 
when  k  is  (/*) ,  and  V  the  number  taken  when  k  is  C"  + 1) ,  we  have 

,  _  1.3.5.7 |>+1]  —  1         ,,_  1.3.5 [2.M  +  1]  [2.U  +  3]  — 1 

a_  g  '      A~  2 


Hence  a'  =  a  +  |>  +  1]  (u) 


23 


To  the  integral  J\  add  and  subtract 

IT 
J     TT  J 

We  get  readily 


(a) 


J\ 


=  **[£ 


-1' coajY^Or—lM^:  --  f(A~1)c°8  C0»)  r  +  ^rM 


(A) 


r; 


(A» 


]• 


Integrating  by  parts  we  get 


(A) 


0«)  +  WJ_i 


•(A-D  sin[(//)r  +  (^)r]c?n  <      (32) 

7T  /  J 


In  particular, 


(A) 


Employing  the  theorem  of  means,  the  second  integral  of  J^  is  seen  to  be  less 
than 


20") 


[-±ri)=M^-^=^-^~)> 


<*0 


2^  + 


which  tends  to  zero  when  //  increases  indefinitely,  since  lim  c^z^O.     Conse- 
quently  lim  «7M  =  lim  |  cM  log  [2/j  -f  1] . 

/a  =<x  /a  =00 

Applying  the  theorem  of  means  to  (32)  we  have 


(A) 


Hence, 


<  2*  C*  [(«)  -  (*)  +  W  +  (/) J  W <  jgj  °A  J0J  ' 


A,  +  1 
2       ^A 


Aj  +  1 
<      2 


Ca 


W  0") 


24 

But  this  series  evidently  has  zero  for  its  limit,  as  is  seen  at  once  by  writing  it 
in  the  form 

VrJ | V^2 

2<"  +  1  ~  oArr       [>  +  l]l>-  1]  - 


[2/i  +  l][2/*  — 1] 


J>) &  +  !)' 

(A+l)  (a<) 

But 

*>  +i 

«^=  «^M  +  1  +  «^»  +       2       «A 
ft  — 1 

and  therefore 

lim   J=  lim   J  cM  log  [2//  +  1]  . 

Now  since  the  c's  are  unrestricted,  except  that  they  are  positive  and  lim  cM  =  0, 

fl=0O 

it  is  possible  to  choose  them  in  such  a  way  that  the  above  product  shall  become 
infinitely  great  with  ft.     This  will  be  the  case  for  example  if  we  take 

c  1 

C*       VlogO  +  1]' 

It  follows,  therefore,  that  with  such  a  choice  of  the  c's,  the  Fourier  series,  which 
represents  the  function  f  (/3)  for  values  of  /9  different  from  zero,  diverges  for 
0=0. 

8.  Let  us  determine  as  far  as  possible  the  nature  of  the  convergence  of  the 
trigonometric  series 

2   (««  sin  nx  -f-  bn  cos  nx) 

n=0 

which  represents  the  function  f(x),  say,  in  the  interval  ( — nt  7c).     Now 

1  fV,  \   .  ,  1  ["/(«)  cos  nay     .      1  f*       ,  , 

an  =  —  /  a  sinwaaaz: k_ i_ i -J f  (a)  cos  na  da 

7i  )-„  '  tc  L  n         _]_„.       n>Tjl.wv  ; 

= tZ^1  [/(?r)  ~/(-  ^~  4  fe  (")  Sin  na da> 

If*                                     If" 
6n  =  —  y  (a)  cos  nada  = f  (a)  sin  na  da 

71  j  —  „.  nTc  J  _  „• 

=  ^^  [/'  (*)  — /'  (—  *)]  +  i  f /"  (a)  cos  na  da . 

This  shows  that,  if  the  first  and  second  derivatives  of/* are  finite,  the  wth  term 
is  of  the  order — ,  except  when  f  ( —  7t)=f(7i),  and  hence   that  in  general 


25 


the  series  is  only  semi-convergent.  But  if,/"( — 7r)=.f(n),  or  if  the  develop- 
ment contains  only  terms  of  the  form  bn  cos  nx ,  the  series  is  absolutely  con- 
vergent. 

The  question  also  arises  :  Is  the  series  uniformly  convergent  ?  Manifestly, 
it  cannot  be  so  in  any  interval  containing  a  point  of  discontinuity.  Let  us  then 
consider  the  question  for  any  interval  (a,  6),  comprised  within  ( — Tt,  tz),  con- 
taining no  point  of  discontinuity  for  the  function,  and  over  which  the  function 
has  not  an  infinite  number  of  maxima  and  minima.  We  need  to  show  that  we 
can  take  n  so  large  that  the  sum  of  the  first  n  terms  of  the  series 


#«  = 


1   f*(ir~a?)/(a?+2y)  sin  kydy 
Tt  Jo  sin  7- 


+ 


sin  y 


[Jfe=2n+1] 


A  = 


shall,  for  any  value  of  x  within  the  interval  (a,  6),  differ  from,/ (x)  by  a  quan- 
tity whose  modulus  is  less  than  a,  where  a  is  an  arbitrary  small  quantity 
chosen  in  advance.     Let  us  put 

x  Jo  sin?"  Tt  J0  sin?- 

f{x)  under  the  conditions  mentioned  above  evidently  possesses  the  property  of 
limited  oscillation  in  the  interval  (a,  6),  and  hence  we  can  write 

f(x  +  2y)  =/(*)  +  P; - JV;',    /(*- 2y)  =zf(x)  +  P>' - N<y>, 
as  long  as  x  -f-  2y  and  x  —  2y  do  not  pass  outside  the  interval  (a,  b).     Or  we 
can  write 

f(x  +  2y)  +/(*-  2y)  =  2f(x)  +  P  -  Ny  . 

Consequently  we  have 


A  = 


Xyf^dr. 


smy 


Since  Py  and  Ny  are  positive  increasing  functions,  we  get,  employing  Bonnet's 
theorem, 


= -§-/(*>! 


rfnfy  ,         P. 

o   sm  y  Tt 


e  sin  ky 
f  sinf 


dy 


Ne   fe  sin  Jcy 


Ne  psi 

Tt      L  S 


?  sm  y 


dy 


ro<*<n 

L0<£'<eJ- 


But  from  (26)  and  (27),  page  17,  we  must  have 


< 


e  smky  7      .  n   ,     2 


j  smy 
Hence  ^L  will  differ  from 


-<f 


sin  ky 


$ '  sin  y 


rfr<^  + 


2  P 


sin  ky 


dr=/{x) 


f/W 


jo  smr 

by  a  quantity  which  is  less  in  absolute  value  than 

4 


sin  £7- 
sin  7" 


dy 


M(JL  + 

~\2^ 


<M, 


26 

where  M  is  the  largest  of  the  quantities  Pe  and  Nt  .  But  since/  (x)  satisfies 
Dirichlet's  conditions,  we  can  choose  £  sufficiently  small  to  have  M<^\a. 
Having  thus  chosen  £,  it  follows,  from  the  theorem  proved  in  §4,  that  we  can 

choose  n  =z  — —  so  large  that 

JL  fix)  F^fr  dr     JL f* (7r~z)/(^+2r)sin^^r 
n  J     JJt    siuy     "      -Je  sinr  ' 

l  ■f*c+")/(g--2r)BiiiJbrdr     (33) 

x  Je  sin?- 

shall,  for  any  value  of  x  in  the  interval  (a,  b),  each  be  less  in  absolute  value 
than  I  a,  and  consequently  we  will  have  Sn  differing  from /(a:)  by  a  quantity 
whose  modulus  is  less  than  a. 

Suppose  now  that  f{x)  has  an  infinite  number  of  maxima  and  minima  in 
the  interval  (a,  6),  but  satisfies  the  condition 

limp  I /(»*»)-»)     dr  =  0.  (28) 

We  can  write  A ,  defined  as  above,  in  the  form 

SlD  kr  dr  4     1  P  ^  +  2^  "^frfl  sin  ^  c?r 
sin;-    '  ~*     7i  J0  siny 

i  f  r./>)-/(*-2r)]sinfcr  d 

7T  J0  sin  y 

Since  the  condition  (28)  is  satisfied,  it  is  easily  seen,  by  employing  the  process 
used  at  the  beginning  of  §6,  that  we  can  choose  £  so  small  that  the  last  two 
terms  in  this  expression  for  A  shall  each  be  less  than  £  a  in  absolute  value. 
Having  thus  chosen  e,  we  can,  as  above,  now  choose  n  sufficiently  large  to  make 
the  modulus  of  each  of  the  expressions  in  (33)  less  than  \o.,  and  hence  we  will 
have  Sn  differing  from  f{x)  by  a  quantity  whose  modulus  is  less  than  a. 

We  have  thus  shown  that  the  Fourier  series  representing  a  function/fa) 
is  uniformly  convergent  in  each  interval  (a,  6),  comprised  within  the  interval 
( — 7r,  tc)  and  containing  no  point  of  discontinuity  for  the  function: 

1°.  When /*(»)  does  not  possess  an  infinite  number  of  maxima  and  minima 
in  this  interval. 

2°.  When  it  fulfils  the  condition 


tM 


lim 


\f(x±2T)-f(x)  l  dr^Q 


Biographical  Sketch. 

Edward  Payson  Manning  was  born  in  Antwerp,  N.  Y.,  March  21st,  1865. 
In  1866  his  parents  removed  to  Raynham,  Mass.,  where  he  lived  until  the  fall 
of  1882,  when  he  entered  the  High  School  of  Providence,  R.  I. 

In  1885  he  entered  Brown  University,  from  which  he  received  the  degree 
of  Bachelor  of  Arts  in  1889. 

After  spending  one  year  in  private  teaching  in  Baltimore,  he  entered  the 
Johns  Hopkins  University  as  a  candidate  for  the  degree  of  Doctor  of  Philo- 
sophy, selecting  Mathematics  as  his  principal  subject,  with  Physics  and 
Astronomy  as  subordinate  subjects.  During  the  last  three  years  he  has  held 
successively  the  positions  of  University  Scholar,  Fellow  and  Fellow  by 
Courtesy,  this  past  year  serving  also  as  an  assistant  in  the  Mathematical 
Department. 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 
BERKELEY 

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REC'l)  LO 
JAN9    1957 

MAR  15  SSI 


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OCT   51960 


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w. 


8 


